Problem: Let represent the operation on two numbers, and , which selects the larger of the two numbers, with . Let represent the operation which selects the smaller of the two numbers, with .
Which of the following three rules is (are) correct?
(1)
(2)
(3)
Answer Choices:
A. (1) only
B. (2) only
C. (1) and (2) only
D. (1) and (3) only
E. all three
Solution:
The correctness of rule is obvious. To establish rule take three numbers such that . First select the larger of and , which is ; then select the larger of and , which is . We thus obtain from the left side of the rule. From the right side of the rule we also obtain as follows: First select the larger of and , which is , then select the larger of and , which is .
For rule we proceed as follows: For the left side we first select the larger of and (this is ). then select the smaller of and (this is ag. in ). For the right side, select the smaller of and (this is ) then select the smaller of and (this is ), and. finally. select the larger of and (this is ).